Definition
Let C be a locally small category and let Set be the category of sets. For each object A of C let Hom(A,–) be the hom functor which maps objects X to the set Hom(A,X).
A functor F : C → Set is said to be representable if it is naturally isomorphic to Hom(A,–) for some object A of C. A representation of F is a pair (A, Φ) where
- Φ : Hom(A,–) → F
is a natural isomorphism.
A contravariant functor G from C to Set is the same thing as a functor G : Cop → Set and is therefore representable just when it is naturally isomorphic to the contravariant hom-functor Hom(–,A) for some object A of C.
Read more about this topic: Representable Functor
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